Coin toss puzzle

This is intended as a counter-example to the view, such as Savage’s, that uncertainty can, in practice, be treated as numeric probability.

You have a coin that you know is fair. A known trickster (me?) shows you what looks like an ordinary coin and offers you a choice of the following bets:

  1. You both toss your own coins. You win if they match, otherwise they win.
  2. They toss their coin while you call ‘heads’ or ‘tails’.

Do you have any preference between the two bets? Why? And …

In each case, what is the probability that their coin will come up heads?

Dave Marsay


In (1) suppose that you can arrange things so that the trickster cannot tell how your coin will land in time to influence their coin, so that the probability of a match is definitely 0.5, with no uncertainty. The situation in (2) can be similar, except that your call replaces the toss of a fair coin.

See Also

Other uncertainty puzzles .

About Dave Marsay
Mathematician with an interest in avoiding 'bad' reasoning.

14 Responses to Coin toss puzzle

  1. Anon says:

    They are equally preferable.

    The difference between the two options is that with (2) it is your decision to call heads or tails. With (1) you are, in effect, tossing your own (fair) coin to make the decision for you. You have no information to determine whether the trickster’s coin is biased to heads or tails, or even if it is biased at all. So there is no reason to call heads rather than tails, or vice versa, so you don’t gain anything by having the freedom choose with option (2).

    • Dave Marsay says:

      Your conclusion is correct according to all the theories of conventional (numeric) probability that I know of. But it seems to me to avoid their assumptions, leaving the issue open. Or have I overlooked a valid theory that deals with this example? Or even a thorough reasoned discussion?

  2. Simon says:

    This is really interesting. Are we able to ask about the trick coin? For example, does the coin enable the trickster to guarantee that it will come out heads or tails, or does the way in which the coin is thrown mean there will be a higher probability of heads or tails? Also, does the trickster have a way of deciding the outcome of his throw after the I have called heads or tails (assuming this call is made while the trickster’s coin is in the air)? Another aspect in the matching scenario would be do I reveal my coin first, and can the trickster then use his ability to change the face of his trick coin to that of mine? issues issues!

  3. Dave Marsay says:

    You can ask, but would you trust the trickster’s reply?

    • Simon says:

      If you look at Anon, they are making an assumption about the nature of the coin, i.e. that it may be biased. But I can think of other tricks, such as the Trickster being able to change the head to a tail once the coin is under their hand. I hope you don’t think i am being too a*** retentive about this! I was thinking about maybe going for option 1 based on the idea that if both coins are revealed at the same time, the trickster is less able to manipulate the coin into showing the side he or she wishes. This maybe ruins any mathematical analysis, as in my head you have to take into account the many ways in which a trickster could cheat.

      • Dave Marsay says:

        The literature is full of assumptions about the situation that anon could be making. It may be that they assume that the trickster is not as clever as they are, and hence that they can model the entire situation probabilistically. If you find yourself unable to make any such assumptions, then Savage recommends a rule which favours (1), as long as you don’t toss your coin first. He gives a good mathematical analysis of this rule, based on considerably less assumptions than anon’s. In my example game theory would also apply, and recommend (1). But neither claims optimality or rules out the possibility of a more precise theory, such as anon claims.

  4. Simon says:

    Thanks Dave. I didn’t want you to think I was being an idiot. I love these kind of puzzles. Some of these kinds of puzzles do mean to find a solution you have to question hidden assumptions.

    • Dave Marsay says:

      The great merit of taking a mathematical approach is that it uncovers those assumptions. The ‘big question’ for my blog is whether these assumptions are at all important in practice, or whether human mistakes have human causes, mainly explainable in terms of psychology etc. For example, many would think that having a choice would always be better than tossing a coin, and hence prefer (2). Is mathematics really helpful in exploring this issue?

  5. Anon says:

    The only assumptions I think (!) I am making are (a) your own coin is fair in option 1 and (b) you have no information regarding whether the trickster’s coin is biased, and if it is whether heads or tails is more likely. (I was also assuming that under option (2) the trickster can’t influence the result of their coin after you have called, but that has been made explicit in the clarification.)

    (b) means that there is no reason to call heads instead of tails in option (2) (or vice versa) so there is no reason to prefer option (2) over option (1).

    Actually, it would be better to say there is no mathematical/financial reason. As David points out some people would choose (2) because it gives them control over the situation (however illusory that control might be). Does that make them irrational? Maybe not if the illusion of control gives them some kind of “utility” (in the economists definition of the word) in addition to the expected financial outcome. On the other hand it may be irrational if they genuinely believe it affects the expected financial outcome (a bit like people who choose lottery numbers, rather than using the option to have them selected at random).

    • Dave Marsay says:

      Anon, you seem to be arguing from first principles rather than invoking any probability theory. Your statement:

      “there is no reason to call heads instead of tails in option (2) (or vice versa) so there is no reason to prefer option (2) over option (1)”

      caught my eye. You seem to be contradicting Savage’s sure-thing principle, which is arguably the corner-stone of his theory of subjective probability.

      Actually, Savage explicitly excludes this type of example from his theory, leaving us to decide the issue for ourselves. And nor do Ramsey et al settle it. Your argument seems to be ‘I can think of no reason to prefer …’ rather than there ‘is’ no reason. I wonder if anyone else can think of a reason? Or demonstrate conclusively that there is none?

  6. Fred says:

    Wouldn’t Occam’s razor suffice as a reason for preferring option (2)?

    • Dave Marsay says:

      An interesting suggestion, thanks. I can’t recall having seen Occam’s razor being used in this type of context, but it does have some appeal. Are there any precedents? (This may be just my ignorance.)

      As a pedantic mathematician, I would say that (1) has a know ‘fair coin’ whereas (2) has what – even to me – is an unknown propensity to call heads or tails. So Occam’s razor seems -to me – to indicate (1). In any case, I would rather trust the coin than my own ability as a randomiser. This may set me apart from most people, who would like the illusion of control that (2) gives.

  7. Fred says:

    No precedents I am aware of. Also, is it a sole flip or a series?

    • Dave Marsay says:

      I was thinking one flip. In a series for (1) the trickster would only learn that the coin really was fair, which doesn’t help ‘him’. In (2) the trickster could learn something about our calls, which may be to his advantage. For example, Shannon found that once people have called a run of Heads they are more likely to call tails, a habit which is exploitable. Even if this wasn’t true, unless I could rule it our for myself, I would go with (1). I would have thought that Occam’s razor would agree, although we are outside the area where I am comfortable with its application.

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